Page:A History Of Mathematical Notations Vol I (1928).djvu/170

In modern notation:

$3x+4$ are worth $8x^{3}+4x^{2}$
how much $8x^{4}-4x$.
Result ${\frac {64x^{7}+32x^{6}-32x^{4}-16x^{3}}{3x+4}}$.

In the treatment of irrationals or numeri surdi Scheubel uses two notations, one of which is the abbreviation Ra. or ra. for radix, or “square root,” ra.cu. for “cube root,” ra.ra. for “fourth root.” Confusion from the double use of ra. (to signify “root” and also to signify x) is avoided by the following implied understanding: If ra. is followed by a number, the square root of that number is meant; if ra. is preceded by a number, then ra. stands for x. Thus “8 ra.” means 8x; “ra. 12” means ${\sqrt {12}}$.

Scheubel’s second mode of indicating roots is by Rudolff’s symbols for square, cube, and fourth roots. He makes the following statement (p. 35) which relates to the origin of $\surd$: “Many, however, are in the habit, as well they may, to note the desired roots by their points with a stroke ascending on the right side, and thus they prefix for the square root, where it is needed for any number, the sign : for the cube root, ; and for the fourth root .”^[1] Both systems of notation are used, sometimes even in the same example. Thus, he considers (p. 37) the addition of “ra. 15 ad ra. 17” (i.e., ${\sqrt {15}}+{\sqrt {17}}$) and gives the result “ra.col. $32+\surd 1020$” (i.e., ${\sqrt {32+{\sqrt {1,020}}}}$). The ra.col. (radix collecti) indicates the square root of the binomial. Scheubel uses also the ra.re (radix residui) and radix binomij. For example (p. 55), he writes “ra.re. $\surd 15-\surd 12$” for ${\sqrt {{\sqrt {15}}-{\sqrt {12}}}}$. Scheubel suggests a third notation for irrationals (p. 35), of which he makes no further use, namely, radix se. for “cube root,” the abbreviation for secundae quantitatis radix.

The algebraic part of Scheubel’s book of 1550 was reprinted in 1551 in Paris, under the title Algebrae compendiosa facilisque descriptio.^[2]

1. “Solent tamen multi, et bene etiam, has desideratas radices, suis punctis cum lines quadam à dextro latere ascendente, notare, atque sic pro radice quidem quadrata, ubi haec in aliquo numero desideratur, notam : pro cubica uerò, : ac radicis radice deinde, praeponunt,”
2. Our information on the 1551 publication is drawn from H. Staigmüller, “Johannes Scheubel, ein deutscher Algebraiker des XVI. Jahrhunderts,” Abhandlungen zur Geschichte der Mathematik, Vol. IX (Leipzig, 1899), p. 431–69; A. Witting and M. Gebhardt, Beispiele zur Geschichte der Mathematik, II. Teil